My elderly sister pays me for assistance. Can she claim this expense on taxes via 1099-Misc? Do I have to claim it as income?

Take a look at IRS Publication 926 Household Employer's Tax Guide

My understanding is that you have to be operating a trade or business to claim expenses on 1099-Misc.  This from the IRS website:

"If you pay independent contractors, you may have to file Form 1099-MISC, Miscellaneous Income, to report payments for services performed for your trade or business. If the following four conditions are met, you must generally report a payment as nonemployee compensation.

You made the payment to someone who is not your employee;
You made the payment for services in the course of your trade or business (including government agencies and nonprofit organizations);
You made the payment to an individual, partnership, estate, or in some cases, a corporation; and
You made payments to the payee of at least $600 during the year"